Compactification of M Theory and Manifolds with G2 Holonomy

本文利用chen - simons理论的连杆不变量，应用几何转换方法，计算了拓扑弦理论在各种非紧Calabi-Yau流形上的配分函数。我们证明了计算结果与Nekrasov对N=2超对称规范理论提出的公式完全一致。从而建立了拓扑弦与N=2 SUSY规范理论之间的关系。本文利用N=2超形式代数的表示理论和模自举方法，导出了N=2 Liouville理论的边界态。已知N=2 Liouville理论是协集SL(2；R)/U(1)模型的t对偶，我们关于边界态的结果与SL(2；R)/U(1)理论的结果非常吻合。在论文7中，我们利用Ashoke-Douglas公式计算了IIB型弦理论中通量真空的分布，并计算了Calabi-Yau流形模空间上通量真空的分布函数。我们证明了分布函数在Calabi-Yau模空间的奇点附近有峰值，并且具有1/(z^2 log z^2)的通用性。因此，它在奇点z=0处发散，但在z=0处仍然可积，并且在每个Calabi-Yau奇点周围只存在有限个真空。在论文8中，我们研究了ALE空间的非紧化Calabi-Yau流形的几何，并给出了它们的计算公式。基于N=2,4超共形代数表示理论分析的椭圆属。我们的公式与利用非平凡的函数恒等式对K3曲面进行反化所得到的公式一致。

In paper 2 we have applied the method of geometrical transition and computed the partition function of topological string theory on various non-compact Calabi-Yau manifolds using the link invariants of Chern-Simons theory. We have shown that the results of computation agree exactly with the formula proposed by Nekrasov for the N=2 supersymmetric gauge theory. And thus they establish the relation between topological string and N=2 SUSY gauge theory.In paper 3 we have used the representation theory of N=2 supersonformal algebra and the method of modular bootstrap and derived the boundary states of N=2 Liouville theory. N=2 Liouville theory is known to be T-dual to the coset SL(2;R)/U(1) model and our results on boundary states agree well with those known in SL(2;R)/U(1) theory.In paper 7 we have used the formula of Ashoke-Douglas for the distribution of flux vacua in type IIB string theory and evaluated the distribution function of flux vacua on the moduli space of Calabi-Yau manifolds. We have shown that the distribution function is peaked around the singular points in Calabi-Yau moduli space and have the universal behavior 1/(z^2 log z^2). Thus it diverges at the singular point z=0, however, it is still integrable around z=0 and there exist only a finite number of vacua around each Calabi-Yau singularity..In paper 8 we have studied the geometry of non-compact Calabi-Yau manifolds like ALE spaces and have proposed formula for their. elliptic genera based on the analysis of the representation theory of N=2,4 superconformal algebras. Our formula agrees with the one suggested from the decompactification of K3 surface by means of some non-trivial theta function identity.